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Complete Entry Pooling Large Support Partial

Econometric Analysis of Games 1

Abi Adams

HT 2017

Abi Adams

TBEA


RecapAim: provide an introduction to incomplete models and partialidentification in the context of discrete games

1. Coherence & Completeness

2. Basic Framework

3. Pooling Outcomes

4. Large Support

5. Partial Identification

6. Adding Structure

Abi Adams

TBEA


Recap

I Last lecture discussed the rank and order conditions thatcharacterise when linear simultaneous systems are pointidentified.

I This week, we will review identification results in theeconometric analysis of discrete games, which are alsosimultaneous systems

I The existence of multiple equilibria results in additionalidentification issues

I After discussing games of complete information, we willreturn to multiplicity in the context of social interactions

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Completeness & CoherenceI Structural model: a set of structural equations and a

distribution between observed and unobservedexplanatory variables specified by an economic theory

h(y ,X , ε) = 0 (1)

with ε ∈ Ω

I The model is coherent if for each ε ∈ Ω there exists atleast one value of y that satisfies the structural equations

I The model is complete if for each ε ∈ Ω there exists atmost one value of y that satisfies the structural equations

I With a complete and coherent model, a unique reducedform is guaranteed

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Completeness & Coherence

I Incoherent and incomplete models can arise insimultaneous systems

I We start with a simple abstract example, and then willconsider these issues in the context of an entry game

I Consider the following simultaneous system

y1 = I(y2 + ε1 ≥ 0)

y2 = θy1 + ε2(2)

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I We have that

y1 = I(θy1 + ε1 + ε2 ≥ 0) (3)

I With unrestricted errors, the model is incomplete: let−θ ≤ ε1 + ε2 < 0:

y1 = I(θy1 + ε1 + ε2 ≥ 0)

ε1 + ε2 0→ y1 = 0θ + ε1 + ε2 ≥ 0→ y1 = 1

(4)

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I With unrestricted errors, the model is incoherent: let0 ≤ ε1 + ε2 < −θ:

y1 = I(θy1 + ε1 + ε2 ≥ 0)

ε1 + ε2 ≥ 0→ y1 6= 0θ + ε1 + ε2 < 0→ y1 6= 1

(5)

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Outline


2. Basic Framework

3. Pooling Outcomes

4. Large Support


6. Adding Structure

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Entry Games

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Entry Games

I Large literature on the estimation on binary games in theempirical IO set-up

I How to measure the competitiveness of a market and thetoughness of competition?

I Bresnahan & Reiss (1991): don’t need toprice/quantity/cost data but can infer from information onequilibrium market structure

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Entry Games: Data

I A firm’s decision to enter depends on its profit, whichdepends on whether the other firm entered the market

I Let profits be given by:

πi = xiβ +4iyj + εi (6)

for i = 1,2

I The distribution of ε ≡ (ε1, ε2) is given by (known) F , withmean normalised to zero and variances to 1

I Start by assuming complete information: realisations of xand ε observed by all players

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Entry Games: Data

I Similar set-up to Ciliberto & Tamer’s (2009) analysis of theairline industry

I Each market is a particular route, the firms being differentairlines

I xi gives the various market and firm specific variables thataffect demand (e.g. population size, income &c) and costsfor firms

I Linearity relaxed in some specifications, but additiveseparability of observables and unobservables typicallyassumed

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Payoff Matrix

y?i = xiβ +4iyj + εi

yi = I(y?i ≥ 0)(7)

Player 20 1

Player 1 0 0, 0 0, x2β2 + ε21 x1β1 + ε1, 0 x1β1 +41 + ε1, x2β2 +42 + ε2

I Nash equilibrium concept

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Payoff Matrix: 4i < 0

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Incomplete

I For certain values of the error term, the model predicts twopossible solutions

I Absent a rule for equilibrium selection, the model isincomplete — cannot write down uniquely determinedlikelihood function because we cannot assign separateprobabilities to (0,1) and (1,0)

I Structural parameters often under-identified from entrydata

I An equilibrium selection mechanism, λ, specifies whichequilibria is selected with which probability

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Incomplete

I How can we deal with this?

I Find a coarser unique prediction — e.g. focus on number ofentrants — useful in a symmetric world but doesn’t alwayswork (Bresnahan and Reiss, 1991; Berry, 1992)

I Exclusion & large support assumptions (Tamer, 2003)

I Complete the model by assuming an equilibrium selectionrule or impose more structure on the game to obtain aunique equilibrium

I Sacrifice point estimates and only find bounds onparameters

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Outline


2. Basic Framework

3. Pooling Outcomes

4. Large Support


6. Adding Structure

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Pooling Multiple Equilibria Outcomes

I A common strategy taken is to aggregate the monopolyoutcomes

I Transform model from one that explains each entrantsstrategy to one that predicts the number of entrants:N = 0,1,2

I Essentially turns the problem into an ordered discretechoice model

I Multiplicity not necessarily an impediment to analysis —here some quantities of interest are invariant acrossequilibria

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I Likelihood function would then include the probabilitystatements

Pr(N = 0|x) = Pr(x1β1 + ε1 < 0, x2β2 + ε2)

= Fε1,ε2(−x1β1,−x2β2)

Pr(N = 2|x) = Pr(x1β1 +41 + ε1 < 0, x2β2 +42 + ε2)

= Fε1,ε2(−x1β1,−x2β2)

Pr(N = 1|x) = 1− Pr(N = 0|x)− Pr(N = 2|x)(8)

I Joint distribution of 4i determines the specific functionalform for these probability statements

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I However, lose information about firm heterogeneitiesunless firms are symmetric

I We then have

Pr(N = 0|x) = Pr(ε1 < −xβ)

Pr(N = 1|x) = Pr(−xβ < ε < −xβ −4)

Pr(N = 2|x) = Pr(−xβ −4 < ε)

(9)

I With ε ∼ N(0,1) we get a very simple ordered probit model

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I While convenient, this strategy imposes strong restrictionson firm level heterogeneity

I Typically want to allow for firms to asymmetric impacts onone another

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Outline


2. Basic Framework

3. Pooling Outcomes

4. Large Support


6. Adding Structure

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Large Support

I Uses similar ideas to those we have discussed in thecontext of special regressors and identification ofsimultaneous systems

I Intuition: find a regressor for which the actions of all butone player are dictated by dominant strategies — turn theproblem into a discrete choice problem by the single agentwho does not play dominant strategies

I For i = 1 or i = 2, there exists a regressor with βik 6= 0 thatis excluded from the other firm’s covariates and haseverywhere positive density

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Large Support

I Point identification of (β1, β2,41,42) then guaranteed if x1and x2 have full column rank

I Can find extreme values fo xik such that only uniqueequilibria in pure strategies are realised, e.g. asx1k → −∞:

Pr(ε1 < −x1β)→ 1Pr(ε1 < −x1 −41β)→ 1

(10)

I Let x?2 be such that:

x?2β2 6= x?2 b2 (11)

which is guaranteed by the full rank condition on x2

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Large Support

I We then have that, as x1k →∞

Pr ((0,0)|x) = Pr(ε1 < −x1β1, ε2 < −x?2β2)

≈ Pr(ε2 < −x?2β2)

6≈ Pr(ε2 < −x?2 b2)

(12)

which implies that β2 is identified

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Large SupportI Let x?1 be such that:

x?1β1 6= x?1 b1 (13)

which is guaranteed by the full rank condition on x1

I We then have that

Pr ((0,0)|x) = Pr(ε1 < −x?1β1, ε2 < −x2β2)

6= Pr(ε1 < −x?1 b1, ε2 < −x2β2)(14)

which implies that β1 is identified

I Could also introduce a separate large support regressorfor firm 2

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Identification at Infinity

I This strategy is often called identification at infinity —using independent variation in one regressor while drivinganother to take extreme values on its support identifies theparameters

I Used in many different applications — Heckman (1990) &c

I However, note that there are important implications forinference: this will lead to asymptotic convergence ratesthat are slower than parametric rates as the sample size(i.e. number of games) increases (see Kahn & Tamer(2010); Bajari et al (2011))

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Outline


2. Basic Framework

3. Pooling Outcomes

4. Large Support


6. Adding Structure

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Partial Identification

I Typically covariates’ support is not rich enough to admit astrategy based on Tamer (2003)

I Can instead rely on partial information and use bounds forestimation — while a model may not make exactpredictions, it may still meaningfully restrict the range ofpossible outcomes

I Advocated by, e.g., Manski (1995)

I These results can be useful for testing particularhypotheses or illustrating the range of possible outcomes

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I Given that the model is incomplete, the probability ofoutcome (0,1) and (1,0) cannot be written as a function ofthe structural parameters

I The model instead provides upper and lower bounds onthese probabilities

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Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(15)

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Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(16)

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Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(17)

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Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(18)

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Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(19)

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Bounds

Pr((0,1)|x) ≤ PU,01(β,4)

= Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 < ε2 < −x2β2 −42

)+ PL,01

(20)

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Bounds

Pr((0,1)|x) ≤ PU,01(β,4)

= Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 < ε2 < −x2β2 −42

)+ PL,01

(21)

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I Can derive similar expressions for the probabilities of otheroutcomes

I The information about the parameters can then berepresented as:

PL,00(θ)PL,01(θ)PL,10(θ)PL,11(θ)

≤

Pr((0,0)|x)Pr((0,1)|x)Pr((1,0)|x)Pr((1,1)|x)

≤

PU,00(θ)PU,01(θ)PU,10(θ)PU,11(θ)

(22)

I The identified set, ΘI , is the set of all parameters thatsatisfy these inequalities

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Example: Ciliberto & Tamer (2009)

I Interested in measuring competitive pressures in the airlineindustry, i.e. the impact on profit of a firm entering aparticular market

I Want to allow for heterogeneity across firms — i.e. theentry decision of American Airlines has a different effect onthe profit of its competitors than the entry of a low costairline

I Some important policy questions regards rules on airportpresence on number of routes served &c

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Example: Ciliberto & Tamer (2009)I Each observation is a route served between two airports

I Cross section study and thus assume firms are in long runequilibrium

I Firm profit function given by:

y?im = Smαi + Zimβi + Wimγi +∑j 6=i

θij yjm +

∑j 6=i

Zjmθij yjm + εim

(23)

I S is a vector of common market characteristicsI Z is a matrix of firm characteristics that enter into the profit

functions of all firmsI W are firm specific characteristics such as cost variables

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I Have the (more complicated!) set of moment equalities PL,1(θ)...

PL,2N (θ)

≤ Pr(y1|x)

...Pr(y2N |x)

≤

PU,1(θ)...

PU,2N (θ)

(24)

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I The estimator uses the following objective function

Q(θ) =

∫ [|| (Pr(y |X )− PL(X , θ))− ||

+|| (Pr(y |X )− PH(X , θ))+ ||]

dFx

(25)

where

(A)− =

a11(a1 ≤ 0)...

a2N 1(a2N ≤ 0)

(26)

and (A+) defined similarly

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I Note that Q(θ) ≥ 0 and Q(θ) = 0 iff θ ∈ ΘI

I To estimate ΘI , take the sample analog of Q(·)

QN(θ) =1N

∑m

[|| (Pn(Xm)− PL(Xm, θ))− ||

+|| (Pn(Xm)− PH(Xm, θ))+ ||] (27)

where Pn(Xm) can be estimated non-parametrically and PLand PH are computed via simulation

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I Unless the number of firms is very small, the upper andlower bound probabilities do not have a convenient closedform

I Common problem with more complicated models:likelihood or moment inequalities do not have a closedform that can adapt standard Maximum Likelihood or GMMmethods for

I Proceed by simulation: calculate an upper and lowerbound for each equilibrium probability for every X for aparticular guess of the parameter values

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Simulation ProcedureI 1. Draw R simulations of the firm unobservables, εr —

these draws and stored and held fixed throughout theoptimisation procedure

I NB can allow for correlation in these draws if each firm’sdraw is random normal by transforming the error termsusing the Cholesky Decomposition of the covariance matrix

I 2. For a given X , a particular draw of the error term, εr andan initial guess of the parameter vector, θ, calculate thevector of firm’s profits for a particular set of entry decisions,y j

I 3. If each firm is earning non-negative profits, then theoutcome y j is an equilibrium

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Simulation Procedure

I 4. If this equilibrium is unique (for no other y j is it true thatall firms profits are positive), then add 1

R to the lower andupper bound for this outcome

I 5. If this equilibrium is not unique, then add 1R to the upper

bound only

I 6. Repeat steps 1-5 for each Xm and εr , r = 1, ...,R

I 7. Calculate the objective simulation estimates of PL andPH

I 8. Repeat steps 1-7 as search for the value of θ thatminimises QN

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Data

I 2001 Airline Origin and Destination Survey

I Markets defined as trips between airports — data setincludes 2742 marets

I Focus on American, Delta, United, Southwest, ‘Medium’Airlines and Low Cost Carriers

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Results Summary

I Heterogeneity in profit functions

I Competitive effects of large and low-cost airlines aredifferent

I Competitive effects of an airline increasing in its airportpresence

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Results

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Conclusion

I To finish, note that many other economic models withsimilar structures

I (Especially!) with strategic interactions, need to consider ifyour model is complete and consider the implications foridentification

I Identification can be achieved through a number ofstrategies; which one is preferable will depend on thecontext and data to hand

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